A counting argument for the geometric Bombieri-Lang conjecture on ramified covers of abelian varieties

TL;DR

This paper proves the geometric Bombieri-Lang conjecture for certain projective varieties over function fields, extending previous results by introducing a new counting argument and advanced algebraic geometry tools.

Contribution

It generalizes prior work by removing hyperbolicity and non-isotriviality assumptions using a novel counting approach and techniques from algebraic geometry and Nevanlinna theory.

Findings

Proves the conjecture for varieties with finite maps to abelian varieties

Introduces a new counting argument for entire curves

Overcomes technical difficulties with advanced geometric tools

Abstract

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the hyperbolicity assumption or the non-isotriviality assumption. The proof builds upon their strategy for constructing entire curves, yet hinges crucially on a new counting argument and draws substantially on tools from algebraic geometry and Nevanlinna theory to overcome various technical difficulties.